The knowledge at which geometry aims is the knowledge of the eternal.

— PLATO, REPUBLIC. VII: 527

Before attempting this unit it is essential that students are given time to revisit the Knowledge Questions in Ideal gas law compared to Euler’s relation in Reason as a Way of Knowing. Here is the culminating graphic juxtaposing a William Blake poetry couplet, the Ideal Gas Law and Euler's relation.

Students should also keep in mind the content and activities in This statement is false, which sabotage absolute certainty and a rock solid foundation to all of mathematics.

FIVE TEXTS FOR THE WRITTEN ASSIGNMENT

All texts from Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.

Right:REUBEN HERSH Professor Emeritus from the Department of Mathematics and Statistics at the University of New Mexico.

TEXT #1: DAVIS AND HERSH ON PLATONISM (318)

According to Platonism, mathematical objects are real. Their existence is an objective fact quite independent of our knowledge of them… They exist outside the space and time of physical existence. They are immutable—they were not created, and they will not change or disappear. Any meaningful question about a mathematical object has a definite answer, whether we are able to determine it or not. According to Platonism, a mathematician is an empirical scientist like a geologist; he cannot invent anything, because it is all there already. All he can do is discover.

TEXT #2: DAVIS AND HERSH ON FORMALISM (339-40)

Mathematics from arithmetic on up is just a game of logical deduction. The formalist defines mathematics as the science of rigorous proof… either we have a proof or we have nothing. Any logical proof must have a starting point. So, a mathematician must start with some undefined terms. These are called “assumptions” or “axioms”… So far as pure mathematics is concerned, the interpretations we give to the axioms is irrelevant. We are concerned only with valid logical deductions from them...

One cannot assert that a theorem is true, any more than one can assert that the axioms are true… Thus the statements of mathematical theorems have no content at all; they are not about anything. On the other hand, according to the formalist, they are free of any possible doubt or error, because the process of rigorous proof and deduction leaves no gaps or loopholes.

World 1 is the physical world, the world of mass, and energy, of stars and rocks, blood and bone. The World of consciousness emerges from the material world in the course of biological evolution. Thoughts, emotions, awareness are non-physical realities. Their existence is inseparable from that of the living organism, but they are different in kind from the phenomena of physiology and anatomy; they have to be understood on a different level. They belong to World 2.

In the further course of evolution, there appear social consciousness, traditions, language, theories, social institutions, all the non-material culture of mankind. Their existence is inseparable from the individual consciousness of the members of the society. But they are different in kind from the phenomena of individual consciousness. They have to be understood on a different level. They belong to World 3. Of course this is the world where mathematics is located.

TEXT #4: DAVIS AND HERSH ON WHAT REAL MATHEMATICIANS DO ACCORDING TO LAKATOS (354)

The actual situation is this… we have real mathematics, with proofs which are established by “consensus of the qualified.” A real proof is not checkable by a machine, or even by any mathematician not privy to the gestalt, the mode of thought of the particular field of mathematics in which the proof is located. Even to the “qualified reader,” there are normally differences of opinion as to whether a real proof (i.e., one that is actually spoken or written down) is complete or correct. These doubts are resolved by communication and explanation, never by transcribing the proof into first-order predicate calculus…

If a theorem is widely known and used, its proof frequently studied, if alternative proofs are invented, if it has known applications and generalizations and is analogous to known results in related areas, then it comes to be “regarded as rock bottom.” In this way of course, all of arithmetic and Euclidean geometry are rock bottom.

TEXT #5: DAVIS AND HERSH ON HUMANIST MATH (410-11)

Mathematics is not the study of an ideal, preexisting non-temporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather it is the part of human studies which is capable of achieving science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact not a question. This fact means no more and no less than the existence of modes of reasoning about ideas which are compelling and conclusive, “noncontroversial when once understood.”

Mathematics does have a subject matter, and its statements are meaningful. The meaning, however, is to be found in the shared understanding of human beings, not in an external nonhuman reality. In this respect, mathematics is similar to an ideology, a religion, or an art form; it deals with human meanings, and is intelligible only within the context of culture. In other words, mathematics is a humanistic study. It is one of the humanities.

The special feature of mathematics that distinguishes it from other humanities is its science-like quality. Its conclusions are compelling, like the conclusions of natural science. They are not simply products of opinion, and not subject to permanent disagreement like the ideas of literary criticism.

As mathematicians we know that we invent ideal objects, and then try to discover the facts about them. Any philosophy which cannot accommodate this knowledge is too small. We need not retreat to formalism when attacked by philosophers. Neither do we have to admit that our belief in the objectivity of mathematical truth is Platonic in the sense of requiring an ideal reality apart from human thought. Lakatos’ and Popper’s work shows that modern philosophy is capable of accepting the truth of mathematical experience. This means accepting the legitimacy of mathematics as it is: fallible correctible, and meaningful

Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.

QUESTIONS FOR THE WRITTEN ASSIGNMENT

1. With reference to the Davis and Hersh texts, outline both the Platonism and Formalism perspectives, especially with regard to whether mathematics is invented or discovered. To what extent do Platonism and Formalism provide a rock solid foundation for mathematics?

2. Although many mathematicians retain vestiges of Platonism and Formalism in their everyday thinking, what are some of the factors that have led to the acceptance of an Humanist approach based on conjectures and refutations.